Quantum walks: the first detected passage time problem

Abstract

Even after decades of research the problem of first passage time statistics
for quantum dynamics remains a challenging
topic of fundamental and practical importance.
Using a projective
measurement approach, with a sampling time τ,
we obtain the statistics
of first detection events for quantum dynamics on a lattice,
with the detector
located at the origin.
A quantum renewal equation for
a first detection wave function, in terms of which the first detection
probability can be calculated,
is derived.
This formula
gives the relation between first detection statistics and the solution
of the corresponding Schrödinger equation in the absence of measurement.
We demonstrate our results with tight binding quantum walk models.
We examine a closed system, i.e. a ring, and reveal
the intricate influence of the sampling time τ
on the statistics of detection,
discussing the quantum Zeno effect, half dark states, revivals
and optimal detection. The initial condition modifies the statistics
of a quantum walk on a finite ring in surprising ways.
In some cases the average detection time is independent of the
sampling time while in others the average exhibits multiple divergences as
the sampling time is modified.
For
an unbounded one dimensional quantum walk the probability of first detection
decays like (time)(−3)
with superimposed oscillations, with
exceptional behavior when the sampling period τ
times the tunnelling
rate γ is a multiple of π/2. The amplitude of the power law decay
is suppressed
as τ→0 due to the Zeno effect.
Our work presented here, is an extended version of Friedman et al.
arXiv:1603.02046 [cond-mat.stat-mech], and it
predicts rich physical behaviors
compared
with classical Brownian motion,
for which the first passage probability density decays
monotonically like (time)−3/2, as elucidated by Schrödinger in 1915.

I Introduction

How long it takes a lion to find its prey,
a particle to reach a domain or
an electric signal to cross a certain threshold?
These are all examples of the first passage time problem
Redner (); Hughes (); MOR (); Benichou ().
A century ago Schrödinger showed that a Brownian
particle in one dimension, i.e. the continuous limit of
the classical random walk, starting at x0, will eventually reach
x=0,
with, however, a probability density function (PDF) of the first arrival
time that is fat tailed, in such a way that the mean first passage time diverges
Schro ().
Ever since, the
classical first passage time has been a well studied field of research.
More recently,
much work has been devoted to the analysis of quantum walks Aharonov1 (); Dorit (); Ambainis (); Childs (); Konno ()
(see Blumen () for a review).
These exhibit interference patterns and ballistic scaling and in that
sense exhibit behaviors drastically different from the classical random
walk. While several variants of quantum walks exist Blumen (),
for example discrete time
walks, coin tossing walks, and tight-binding models,
one line of inquiry addresses a question generally
applicable to all these cases, namely
the statistics of first passage or detection times of a quantum particle
(to be defined precisely below).
Quantum walk search algorithms which are
supposed to perform better than classical walk search methods vitalized
this line of research in recent years.
A physical example might be the statistics
of the time it takes a single electron, ion or atom
to reach a detection
device.
This question, which at first sight appears well-defined and physically
meaningful, has nevertheless been the subject of much controversy.
The Schrödinger Eqn. and the standard
postulates of quantum mechanics CT () do not give a ready-made
recipe for calculating these statistics.
There is no textbook quantum
operators or wave function associated with the first passage time
measurements
(see Hauge (); Muga (); PHD ()
for related historical accounts). Actually, time is a non-quantum ingredient
of quantum mechanics and is treated as a object detached from
the probabilistic interpretation inherent to non-classical reality.
From the non-deterministic nature of quantum mechanics,
we may expect that the time it takes a single particle
to reach a detection point or domain for a given
Hamiltonian and initial condition should be random
even in the absence of external noise,
but how to precisely obtain
the distribution of first detection times has remained in our opinion
a controversial
matter.

The key to the solution is that
we must take into consideration the measurement process
Bach (); Brun06 (); Brun07 (); Brun08 (); Dhar (); Dhar1 ().
For example consider a zebra sitting at the origin waiting for a lion
to arrive for the first and, unfortunately for her, the last time.
At some rate, the zebra
records: did the lion arrive or did it not?
The outcome is a string of answers:
e.g. no, no, no, …. and finally yes.
If the lion is a quantum particle, then continuous attempts to detect it
by the zebra will maintain
the zebra’s life, since the wave function of the lion is collapsed
in the vicinity of the zebra;
this is the famous quantum Zeno effect Misra ()
(see more details below).
On the other hand, if the
zebra samples the arrival of the lion at a finite constant rate,
its likelihood of death is much higher.
In this sense the measurement of the time of first detection,
which implies a
set of null
measurements for times prior to the final positive recording
is very different than the familiar measurements of canonical variables
like position and momentum. There the system is prepared at time t=0 in
some initial state, it evolves free of measurement until time
t, at which point an instantaneous recording of some observable is made.
Furthermore, we must distinguish between first arrival or first
passage problems Muga ()
and first detection at a site.
Note that even classically the first detection does not imply
that the particle arrived at the site for the first time at the moment of
detection if the sampling is not continuous in time.
More importantly, arrival
times are ill-defined in quantum problems Aharonov (), because we
cannot have a complete record of the trajectory of a quantum particle,
whereas the first detection problem under repeated
stroboscopic measurements is a well-defined problem, and that is what we treat in this manuscript.

Here we investigate the first detection problem of quantum walks
following Dhar et al. Dhar (); Dhar1 () who formulated the problem as a tight-binding
quantum walk, with projective local measurements every τ
units of time (see also Brun06 ()).
Specifically, we consider
a particle on a discrete graph, the quantum evolution determined by the time
independent Hamiltonian H. Initially the particle is localized
so the state function is |ψ⟩=|x⟩ (some of our general results
are not limited to this initial condition, see below).
Detection
attempts are performed locally at a site we call the origin which is
denoted with |0⟩. Measurements on the origin are stroboscopic with the sampling time
τ, and as mentioned the measurement stops once the particle
is detected after n attempts, so nτ is the random
first detection time. We investigate
the statistics of the random observable n.
The questions are: is the particle eventually detected?
what is the probability of detection after n attempts?
what is the average of number of attempts of detection before a successful measurement?
This we investigate both for closed systems and open ones.
Below we present a physical derivation of the quantum renewal equation describing
the probability amplitude of first detection for
the transition |x⟩→|0⟩. In classical stochastic theories
this corresponds to Schrödinger’s renewal equation Schro ()
for the first time a particle
starting on x reaches 0 (see details below).
We show
how the solution of Schrödinger’s wave equation free of measurement
can be used to predict the quantum statistics of the first detection time.
Previously Grünbaum et al. Grunbaum () considered the case where the starting
point is also the detection site |x⟩=|0⟩.
A topological interpretation of the detection
process was provided for that initial condition and among other things they showed that
the expected time of first detection is an integer times τ or infinity.
This integer is the winding number of the so called Schur function of
the underlying scalar measure, the latter is determined by the initial state and
the unitary dynamics. Hence the expectation of the first detection time is
quantized Sinkovicz ().
A vastly different behavior is found when we analyse the transition |x⟩→|0⟩ for
x≠0Dhar1 (); Brun06 (). The average of n is not an integer, neither is detection finally
guaranteed.
As demonstrated below for a ring geometry half dark states are observed in some cases
while in others the average of n exhibits divergences and non-analytical behaviours,
for certain critical sampling times. Finally, we show that critical sampling,
including slowing down is found even for an infinite system.
Namely, for the quantum walk on the line,
the first detection probability decays like a power law, with additional
oscillations, where the amplitude of decay is not a continuous function of the sampling rate.
Thus rich physical behaviors are found for the quantum
first detection problem, if compared with the known results of the classical
random walker.

The spatial quantum first detection problem is a timely
subject. Current
day experiments on quantum walks can be used to study these problems
in the laboratory Yaron (); QWe2 (); QWe1 (); QWe (); Xue (). First passage time
statistics in the classical domain
are usually recorded based on single particle analysis.
Namely, one takes, say, a Brownian
particle, releases it from a certain position and then detects its time of arrival at
some other location.
This single particle experiment is repeated many times and then a
histogram of the first passage time is reported. While in principle one
could release simultaneously many particles from the same position,
their mutual interaction will influence the statistics of first arrival
and similarly statistics of identical particles, either bosons or fermions,
alter
the many particle statistics compared to the single particle case.
Hence, measurement
should be made on single particles, or in other words at least classically
the first detection
time is a property of the single particle path and hence its history.
The recent advance
of single particle quantum tracking and measurement,
for systems where coherence is maintained for relatively long times,
is clearly a reason
to be optimistic with respect to possible first detection measurements.
Such measurements could test our predictions as well as those
of
a variety of
other theoretical approaches
Muga (); Aharonov (); Lumpkin (); Grot (); Stefanak (); Shikano (); Krap (); Ranjith (); Das (); Miquel (),
some of which are compared with our results
towards the end of this manuscript.

The navigation map of this manuscript is as follows.
We start with the presentation of the quantum walk model and
the measurement process in Sec. II.
In Sec. III the first detection wave function formalism
is developed.
The main tool for actual solution of the problem is
based on the generating function formalism
given in Sec. IV
and in subsection
IV.1
the quantum renewal equation is discussed.
Sec. V presents the example of first detection on rings,
with special emphasis on the peculiar statistics
found on a benzene-like ring.
In Sec. VI we obtain statistics of first detection times,
for a one-dimensional quantum walk on an infinite lattice.
We end with further discussion of previous results (Sec.
VII)
and a summary.
A short account of part of our main results was recently published
FKB ().

Ii Model and Basic Formalism

We consider a particle whose evolution is described by a time independent
Hermitian Hamiltonian H according to the Schrödinger Eqn.
iℏ|˙ψ⟩=H|ψ⟩.
The initial condition is denoted |ψ(0)⟩.
For simplicity, we consider a discrete x-space.
As an example we shall later consider the tight binding model

H=−γ∞∑x=−∞(|x⟩⟨x+1|+|x+1⟩⟨x|)

(1)

on a lattice, though our general results are not limited to
a specific Hamiltonian.
We denote a subset of lattice points X,
and loosely speaking we are interested in
the statistics of first passage times from the
initial state to any site x∈X in the subset. More generally, X could be any subset of orthogonal states.
An example is when X
consists of a single lattice point, say x=0 and initially the
particle is localized at some other lattice
point |ψ(0)⟩=|x′⟩. We then investigate the
distribution of the first detection times. For that we must define the
measurement process following Dhar (); Bach (); Ambainis ().

Measurements on the subset X are made at discrete times
τ,2τ,⋯,nτ⋯ and hence
clearly the first recorded detection
time is either tf=τ or 2τ etc.
The measurement provides two possible outcomes:
either the particle is in x∈X or it is not.
Consider the first
measurement at time τ. At time τ−=τ−ϵ with
ϵ→0 being positive, the wave function is

|ψ(τ−)⟩=U(τ)|ψ(0)⟩

(2)

and U(τ)=exp(−iHτ/ℏ) as usual. In what follows,
we set ℏ=1.
The probability of finding the
particle in x∈X is, according to the standard interpretation,

P1=∑x∈X|⟨x|ψ(τ−)⟩|2.

(3)

If the outcome of the measurement is positive, namely the particle is found in x∈X,
the first detection time is tf=τ. On the other hand, if the particle
is not detected, which occurs with probability 1−P1,
the evolution of the quantum state will
resume. According to collapse theory, following the measurement
the particle’s wave function in x∈X is zero. Namely, a null
measurement alters the wave function in such a way
that the probability of detecting the
particle in x∈X at time τ+ϵ vanishes.
In this sense we are considering projective measurements whose duration
is very short, while between the measurements the evolution is according to
the Schrödinger Eqn. Mathematically the measurement is a projection CT (),
so that at time τ+=τ+ϵ we have

|ψ(τ+)⟩=N(1−∑x∈X|x⟩⟨x|)|ψ(τ−)⟩,

(4)

where 1 is the identity operator,
and the constant N is determined from the normalization condition.
Here we have used the assumption of a perfect projective measurement
that does not alter either the relative phases
or magnitudes of the wave
function not interacting with the measurement device, i.e.,
outside the observation domain the wave function is left unchanged beyond a global renormalization.
This is the fifth postulate of quantum mechanics CT (),
though clearly it should be the subject to
continuing experimental tests.
Since just prior to
measurement the probability
of finding the particle in x not belonging to X is 1−P1
we get

|ψ(τ+)⟩=1−∑x∈X|x⟩⟨x|√1−P1|ψ(τ−⟩=1−∑x∈X|x⟩⟨x|√1−P1U(τ)|ψ(0)⟩.

(5)

In sum, the measurement nullifies the wave functions on x∈X
but maintains the relative amplitudes of finding the particles outside
the spatial domain of measurement device, modifying only the normalization.

We now proceed in the same way to the second measurement. Between the
first and second detection attempts we have
|ψ(2τ−)⟩=U(τ)|ψ(τ+)⟩.
The probability of finding the particle
in x∈X at the second measurement,
conditioned on the quantum walker not having been
found in the first attempt is

This iteration procedure is continued to find the
probability of first detection in the n-th measurement, conditioned on
prior measurements not having detected the particle

Pn=∑x∈X|⟨x|[U(τ)(1−^D)]n−1U(τ)|ψ(0)⟩|2(1−P1)....(1−Pn−1).

(9)

In the numerator the operator 1−^D appears n−1 times
corresponding to the n−1 prior measurements. Similarly, in the
denominator we find
n−1 probabilities of null measurements 1−Pj.
Following Dhar (); Dhar1 () we define the first detection wave function

(10)

or equivalently
|θn⟩=[U(τ)(1−^D)]n−1|θ1⟩
with the initial condition
|θ1⟩=U(τ)|ψ(0)⟩.
The bra |θn⟩ is defined only for the moments of detection
n=1,2,⋯,
unlike |ψ(t)⟩ which is a function of continuous time.
With this definition

Pn=⟨θn|^D|θn⟩Πn−1j=1(1−Pi).

(11)

The main focus of this work is on the probability of first detection
in the n-th measurement, denoted Fn. This is of course not the same as
Pn which as mentioned is a conditional probability, namely the
probability of detection on
the n-th attempt given no previous detection.
The conceptual measurement process for the calculation
of Fn is as follows. We start with an initial spatial wave function
|ψ(0)⟩ and evolve it until time τ when the
detection of the particle in x∈X is attempted, and with probability
P1 the first measurement is also the first detection. Hence to simulate this process on a
computer we toss a coin using an uniform random number generator and if
the particle is detected the measurement time is τ.
If the particle
is not detected we compute P2.
Then at time 2τ either the particle is
detected with probability P2
or not.
This process is repeated until a measurement is recorded (see remark below)
and that
measurement constitutes the random first detection event.
In order to gain statistics of the first detection time we
return to the initial step and restart the process with the
same initial condition.
In this way, repeating this many times,
we construct the
first detection probability

We see that the first detection probability Fn is the expectation value
of the projection operator ^D
with respect to
|θn⟩, which we term
the first detection wave function.

Remark. We shall see that not all sequences of
measurements, generated on a computer or in the lab,
yield a detection in the long-time limit.
This is not problematic since also
classical random walks in say three dimensions are not recurrent and hence
the total
probability of detection is not necessarily unity.
In many works one defines the survival probability, i.e.,
the probability that
the particle is not detected
in the first n attempts,

Sn=1−n∑n=1Fn.

(14)

The eventual survival probability
S∞
can be equal zero or not. If the initial condition and the detection location
are identical and S∞=0 the quantum walk is called recurrent.
We will later investigate whether or not the
quantum walk is recurrent, both for the cases of an infinite lattice and
a finite ring.

Iii First detection amplitude

In this section, we solve the first detection time problem
for quantum dynamics
with a single detection site, which we label x=0, so
^D=|0⟩⟨0|.
We define the amplitude of the first detection
as

ϕn=⟨0|θn⟩

(15)

so that Fn=|ϕn|2.
Using Eq. (10) ϕ1=⟨0|U(τ)|ψ(0)⟩,
ϕ2=⟨0|U(2τ)|ψ(0)⟩−ϕ1⟨0|U(τ)|0⟩ and a short calculation yields

We call this iteration rule
the quantum renewal equation.
It yields the amplitude ϕn in terms
of a propagation free of measurement; i.e.,
⟨0|U(nτ)|ψ(0)⟩ is the amplitude for being at
the origin at time nτ in the absence of measurements, from which we
subtract n−1 terms related to the previous history of the system.
The physical interpretation of Eq. (17)
is
that the condition of non-detection in previous measurements translates into
subtracting wave sources (hence the minus sign)
at the detection site |0⟩ following the jth detection attempt.
This is due to the nullification of the wave function at the
detection site
in the jth measurement.
The evolution of that wave source from the jth
measurement onward is described by the free Hamiltonian, hence
⟨0|U[(n−j)τ]|0⟩ which gives the amplitude of return back
to the origin, in the time interval (jτ,nτ).

We now consider the formal solution to the first detection problem for
an initial condition on the origin hence |ψ(0)⟩=|0⟩
and
as mentioned the origin is also the point at which we perform the detection trials.
Clearly in this case ϕ1=⟨0|U(τ)|0⟩ and since
U(0)=1 we get ϕ1=1 when τ→0 which is
expected. For ϕ2=⟨0|U(2)|0⟩−⟨0|U(1)0⟩2
where we use the short-hand notation
U(n)≡U(nτ).
Similarly
ϕ3=⟨0|U(3)|0⟩−2⟨0|U(2)|0⟩⟨0|U(1)|0⟩+⟨0|U(1)|0⟩3.
The general solution is obtained by iteration
using Eq.
(17),

ϕn=n∑i=1∑{m1,⋯,mi}(−1)i+1⟨0|U(m1)|0⟩⋯⟨0|U(mi)|0⟩.

(18)

The double sum is over all partitions of n, i.e. all
i-tuples of positive integers
{m1,....mi}
satisfying m1+⋯+mi=n. For example for n=5 we have five
partitions
corresponding to i=1,⋯,5, for i=1 the set in the second sum is
{5}, for i=2 we sum over {2,3},{3,2},{1,4} and {4,1},
for i=3 we use {1,1,3},{1,3,1},{3,1,1},{2,2,1},{2,1,2},{1,2,2},
for i=4, {1,1,1,2},{1,1,2,1},{1,2,1,1},{2,1,1,1}
and for i=5 we have one term {1,1,1,1,1}.
Hence

With a
symbolic program
like Mathematica one can obtain similar exact expressions for intermediate
values of n. However, to gain some insight
we
turn now to the generating function approach Brown ().

Iv Generating function Approach

The Z transform, or discrete Laplace transform,
of ϕn is by definition Brown (); remark1 ()

^ϕ(z)=∞∑n=1znϕn.

(20)

^ϕ(z) is also called the generating function.
Multiplying Eq. (17) by zn and summing over n

^ϕ(z)=∞∑n=1⟨0|znU(n)|ψ(0)⟩−∞∑n=1n−1∑j=1ϕjzj⟨0|zn−jU(n−j)|0⟩.

(21)

Evaluating the first term on the right hand side we get

^U(z)=∞∑n=1znU(n)=∞∑n=1exp(−iHτn)zn=ze−iHτ1−ze−iHτ.

(22)

The second term in Eq. (21) is a convolution term and after
rearrangement we find one of our main results remark1 ()

^ϕ(z)=⟨0|^U(z)|ψ(0)⟩1+⟨0|^U(z)|0⟩

(23)

or more explicitly

^ϕ(z)=⟨0|1z−1eiHτ−1|ψ(0)⟩1+⟨0|1z−1eiHτ−1|0⟩.

(24)

This equation, relates the generating function ^ϕ(z) to
the Hamiltonian
evolution between the initial condition and the detection attempt.

This approach is also valid for
other types of measurements, repeatedly performed at times
τ,2τ,⋯.
For example the case where we measure a set of points x∈X is
given in Eq.
(100)
in Appendix
A.
First detection measurements
of general observables is also treated there.

iv.0.1 Relations between ^ϕ(z) and
ϕn, S∞ and ⟨n⟩.

As usual the amplitudes ϕn are given in terms of their
Z-transforms by the inversion formula

ϕn=1n!dndzn^ϕ(z)|z=0

(25)

or

ϕn=12πi∮C^ϕ(z)z−n−1dz

(26)

where C is a counter clockwise path that contains the origin and is
entirely within the radius of convergence of ^ϕ(z).

The probability of being measured
is also related to the generating function ^ϕ(z) by

1−S∞=∞∑n=1Fn=∞∑n=1|ϕn|2=

12π∫2π0∞∑k=1ϕkeiθk∞∑l=1ϕ∗le−iθldθ=12π∫2π0|^ϕ(eiθ)|2dθ.

(27)

Similarly

⟨n⟩=∞∑n=1nFn=12π∫2π0[^ϕ(eiθ)]∗(−i∂∂θ)^ϕ(eiθ)dθ.

(28)

The latter is the average of n only when the particle
is detected with probability one, namely when S∞=0.
A shorthand notation of Eq. (28)
is
⟨n⟩=⟨^ϕ|−i∂θ|^ϕ⟩.

iv.1 Connection between first detection and spatial wave function

In classical random walk theory the key approach to the first passage time
problem is to relate it to occupation probabilities Redner ().
Let us unravel a similar relation in the quantum domain, connecting
between first detection statistics and the corresponding
wave packet, namely the time dependent
solution of the Schrödinger equation in the absence
of measurement (see also
Grunbaum () for the |0⟩→|0⟩ transition).
To that end, we first briefly review the classical random walk.

Consider a
classical random walk in discrete time
t=0,1,..,
for example
a random walk on a cubic lattice in dimension d
with jumps to nearest neighbours. The main assumption is that
the random walk is Markovian.
Denote Pcl(r,t) as the probability
that the walker
is at r at time t when initially the particle is at the origin
r=0
and in the absence of any absorption.
Let Fcl(r,t) be the first passage probability:
the probability
that the random walk visits site r for the first time
at time t with the same initial condition.
Following the first equation in
the first chapter in Redner (), Pcl(r,t)
and Fcl(r,t) are related
by

Pcl(r,t)=δr0δt0+∑t′≤tFcl(r,t′)Pcl(0,t−t′).

(29)

This equation Schro (); Montroll (); MontrollW (),
sometimes called the renewal equation, is generally valid
for Markov processes in the sense
that it is not limited to discrete time and space models; in the continuum
one needs only to replace summation with integration,
and probabilities by probability densities.
The idea
behind Eq. (29)
is that a particle on position r at time t must have either arrived
there previously at time t′for the first time
and then returned back or it arrived
at r exactly at time t for the first time (the t′=t term)
Schro (); Redner ().
Using the Z transform the following equations are derived Redner ()

Fcl(r,z)=⎧⎪⎨⎪⎩Pcl(r,z)Pcl(0,z)r≠01−1Pcl(0,z)r=0.

(30)

From this formula, various basic properties of random walks
can be derived. One example is
the Pólya theorem which
answers the question: does a particle eventually return to its origin;
i.e., whether the random walk is recurrent.
A second
is that in one dimension, for an open system
without bias, the famous law Fcl(0,t)∼t−3/2 is found
for large first passage time t and hence the first
passage time has an infinite mean, as mentioned in the introduction.
We will later find the quantum analogue to this well known t−3/2 behaviour.

At first glance this classical picture might not seem related to ours.
However consider the case where we detect the particle at the origin,
so ^D=|0⟩⟨0| and initially
|ψ(0)⟩=|0⟩.
Then Eq. (24) reads

By definition the sum
⟨0|∑∞n=0znexp(−iHτn)|0⟩
is the generating
function
of the amplitude of being at the origin
retrieved from the solution of the Schrödinger equation without detection.
Namely, let |ψf(t)⟩ be the solution of the Schrödinger
equation for the same initial condition |ψf(0)⟩=|0⟩
(the subscript f denotes a wave function free of measurement).
The amplitude of being at the
origin at time t is
⟨0|ψf(t)⟩ and
|ψf(t)⟩=exp(−iHt)|0⟩ as usual.
We define the generating function
of this amplitude, for the sequence of measurements under consideration

⟨0|^ψf(z)⟩0≡∞∑n=0zn⟨0|ψf(nτ)⟩

(35)

and clearly ⟨0|^ψf(z)⟩0=∑∞n=0⟨0|znexp(−iHτn)|0⟩, the subscript zero
denotes the initial condition.
Hence we get the appealing result
reported already in Grunbaum ()

^ϕ(z)=1−1⟨0|^ψf(z)⟩0.

(36)

Thus the generating function of the first detection time
is determined from the Z transform
of the spatial wave function at the point of detection x=0.
This connection is the quantum analogue of the second
line in the classical expression Eq. (30)
since in both cases we start and detect at the origin.

Similarly for an initial condition initially localized at some site x≠0,
so |ψ(0)⟩=|x⟩ with detection at site 0 we find

^ϕ(z)=⟨0|^ψf(z)⟩x⟨0|^ψf(z)⟩0

(37)

where |^ψf(z)⟩x is the Z transform of the wave function
free of measurements initially localized on site x,
|^ψf(z)⟩x=∑∞n=0zn|ψf(nτ)⟩x
with
|ψf(nτ)⟩x=exp(−iHnτ)|x⟩.
We see that the ratio of the generating functions of the amplitudes
of finding the particle on |0⟩ for initial condition on x
and the location of measurement site 0,
obtained from the measurement-free evolution, yields the generating
function of the measurement process. This is the sought after quantum renewal
equation, namely the amplitude analogue of the upper line of the classical
Eq.
(30).

Remark
In Eq.
(35)
the lower limit of the sum is n=0, while
in Eq. (20) the sum starts at n=1 as noted already remark1 ().
Since ϕ0=0 one may of course use a summation from 0 also in Eq.
(20).

Remark Our formalism is not limited to spatially
homogeneous Hamiltonians. Note that in our classical discussion,
following the textbook treatment Redner () and
for the sake of simplicity, we have assumed translation
invariant random walks. In non-translation invariant systems,
one should replace Pcl(0,t−t′) in the left hand side
of Eq.
(29)
with
Pcl(r,t−t′|r,0).
Since the convolution structure of the equation remains,
related to the Markovian
hypothesis, Eq.
(30)
can be easily modified to include non-homogeneous effects.

Remark Sinkovicz et al. Sinkovicz1 ()
found a quantum Kac-Lemma for recurrence
time, thus analogies between quantum and classical walks are not limited
to the renewal equation under investigation.

iv.2 Zeno Effect

The amplitude of finding the particle
at the origin in the first attempt, is given by the initial wave function
projected on the origin, i.e. the probability amplitude of finding
the particle at the origin at t=0.
Hence the above expression gives an obvious answer
for the first measurement; the repeated measurements being very frequent
do not allow the wave function to be built up at the origin, and hence
ϕn=0 for all n>1.
This means that we may investigate the problem
for τ small relative to the time scales of the Hamiltonian, but we cannot
take the limit τ→0 if we wish to retain information
on the measurement process beyond
the initial state.

iv.3 Energy representation

so that the operator ^U(z) is diagonal in the energy representation.
Here |Ei⟩ is a stationary state of the Hamiltonian H,
namely H|Ei⟩=Ei|Ei⟩.
Clearly it is worthwhile presenting the solution in that basis.
Consider the example of the measurement at the spatial origin corresponding
to state |0⟩. This state can be expanded in the energy representation
|0⟩=∑kCk|Ek⟩ with Ck=⟨Ek|0⟩.
Here as usual ⟨Em|Ek⟩=δmk.
Similarly the initial condition is expanded
as |ψ(0)⟩=∑kAk|Ek⟩. The matrix
element

⟨0|^U(z)|ψ(0)⟩=∑kC∗kAk[z−1exp(iEkτ)−1]−1

(40)

together with

⟨0|^U(z)|0⟩=∑k|Ck|2[z−1exp(iEkτ)−1]−1

(41)

yields ^ϕ(z) using Eq. (23).
For the special case where |ψ(0)⟩=|0⟩ we get Ak=Ck and

^ϕ(z)=∑k|Ck|2[z−1exp(iEkτ)−1]−11+∑k|Ck|2[z−1exp(iEkτ)−1]−1

(42)

Here as usual ∑k|Ck|2=1.
It is easy to check that when τ→0 we get F1=|ϕ1|2=1
since a particle starting at the origin is with probability one detected when
τ→0.

Figure 1:
Schematic model of a benzene ring.
In the text measurement is performed on site 0 and we discuss
several initial conditions.
Figure 2:
For a quantum walk on a benzene ring, with initial condition
|ψ(0)⟩=|0⟩ and projective measurements on the origin,
the average number of detection attempts is, by Eq. (49), 4 except for
the cases when γτ is a multiple of π/2 or 2π/3.
Here we plot ⟨n⟩N=∑Nn=1nFn for N=500
and N=10000,
the results converging (not shown) as we increase N further.

V Rings

For our explicit calculations, we will focus
on tight binding models in one dimension Blumen (). The first model
is a quantum walk on a ring of length L:

H=−γL−1∑x=0(|x⟩⟨x+1|+|x+1⟩⟨x|).

(43)

This describes a quantum particle jumping between nearest neighbours
on the ring. We use periodic boundary conditions and thus from the site
labeled x=L−1 one may
jump either to the origin x=0 or to the site labeled x=L−2.
In condensed matter physics the parameter γ is called the
tunnelling rate.

v.1 Benzene-Type Ring

As our first example we consider the tight-binding model on a hexagonal ring
presented in Fig.
1,
namely a structure similar to the benzene molecule CT (); Feynman ().
We consider the influence of initial states |ψ(0)⟩=|x⟩ with
x=0,1,..5
on the statistics of first detection times for detection at site 0
so ^D=|0⟩⟨0|.
According to our theory, to find the generating function we need
the energy levels of H and its eigenstates.
The six energy levels of the system
are Ek=−2γcos(θk)
with k=0,...5 and the eigenstates are |Ek⟩T=(1,eiθk,e2iθk,ei3θk,ei4θk,ei5θk)/√6
with θk=2πk/6CT () (T is the transpose).
Hence the coefficients
|Ck|2=|⟨Ek|0⟩|2=1/6, reflecting the symmetry of the problem.

It is interesting to note that
the generating function satisfies the identity

^ϕ(eiθ)^ϕ(e−iθ)=1.

(46)

an identity we will return to below when discussing ⟨n⟩
and S∞.
Inserting Eq. (46) in Eq.
(27) and integrating over θ gives S∞=0.
Thus the survival probability is zero in the long time
limit.
This behavior is classical in the sense that
for finite systems a classical random walker is always detected.
Note that for a quantum walker this conclusion is not generally valid.
If we start at |1⟩ for example and measure at |0⟩,
and perform measurements on full revival periods,
the particle is never recorded (see further details
and other examples below).
Hence, for a quantum particle
the survival
probability Sn does not generally decay to zero as n→∞, even for finite systems.

For special values of γτ we get exceptional
behaviors.
When γτ is 2π times an integer we get
^ϕ(z)=z namely the
measurement in the first attempt is made with probability 1,
so the first detection time is τ, which is expected
since the wave function is fully revived at these τ’s
in its initial state at the origin
Blumen ().
If γτ=π we get ^ϕ(z)=z(3z−1)/(3−z). Inverting
we find ϕ1=−1/3 and ϕn=8/3n for n≥2,
thus the amplitude ϕn decays exponentially.
It follows that the first detection probabilities are

Fn=⎧⎪
⎪
⎪⎨⎪
⎪
⎪⎩19n=1649nn≥2.

(47)

The average number of detection attempts is ∑∞n=1nFn=2.
If γτ=π/2 we find ^ϕ(z)=−z(1+2z+3z2)/(3+2z+z2)
which has
simple poles and hence

Fn=⎧⎪
⎪⎨⎪
⎪⎩1/9n=116/81n=2243nsin2(ζ1(n−2)−ζ2)n≥3

(48)

where
ζ1=tan−1(√2) and
ζ2=tan−1(√2/5).
For this case ⟨n⟩=3. Similarly for γτ=2π/3+2kπ
and γτ=4π/3+2kπ we
get ⟨n⟩=2.
The general feature of finite rings is an exponential decay of
Fn with a superimposed oscillation determined by the poles of
the generating function. However the sampling times
γτ=0,π/2,2π/3,π⋯ considered so far
exhibit behaviors
which are not typical, as we now show.

A surprising behavior is found for the average, with

⟨n⟩=4

(49)

for any sampling rate in the interval (0,2π) besides
what we call the exceptional sampling times
γτ=0,π/2,2π/3,π,⋯ where as mentioned
⟨n⟩=1,3,2,2,⋯ respectively,
which is continued periodically (see Fig. 2). This result is derived below.
As mentioned in the introduction the fact that ⟨n⟩ is some integer was
already pointed out rather generally by Grunbaum () and this is related to topological
effects.
Except for the exceptional points, the variance of n is

Var(n)=−11+274−4cosγτ+16cos2γτ+34+4cosγτ+3(1+2cosγτ)2

(50)

so that the first detection time exhibits large fluctuations near these
points.
Thus for the 0 to 0 transition
it is only the average ⟨n⟩ that is nearly always
not sensitive to the sampling
rate, not the full distribution of first detection times.

There are numerous methods to find ⟨n⟩=∑∞n=1nFn.
For the exceptional points we used the exact solution for Fn
(as mentioned).
For other sampling times we use two approaches: the first using
Mathematica and is based on a Taylor expansion
of ^ϕ(z) and the second is an analytic calculation.
The former approach is very general in the sense that it can be used
in principle for general initial conditions and other problems beyond the benzene ring.

Specifically, we calculate Fn exactly using the expansion
of ^ϕ(z) with
symbolic programming on Mathematica. This is performed
up to some large N. We then calculate ⟨n⟩N=∑Nn=1nFn. Clearly ⟨n⟩>⟨n⟩N, and increasing N
we see convergence towards ⟨n⟩=4 except for the mentioned
exceptional points. An example is shown in Fig.
(2) for the cases N=500 and N=10000.

Even better is to write ^ϕ(z)=z4H(1/z)/H(z), which is the extension to general z of the identity
Eq. (46)
that we used to show S∞=0.
To find ⟨n⟩ we use Eq.
(44) to find

where α (or β) is the number of zeros of H(z) for z within
(or on) the unit
circle respectively. As explained in Appendix B,
α=0 for otherwise we would find Fn>1.
For the exceptional values of γτ we find β>0, as follows:

This agrees with the values of ⟨n⟩
we have found at the exceptional
points. This exercise shows that mathematically, at least for
this example, the exceptional
points are those specific
values of τ where some of the zeros of the polynomial H(z) are found to lie on the unit circle in the complex plane.
We will soon find a by far more physical
and explicit formula for these points, Eq. (58) below.

x

0<γτ<2π∗

γτ=0

12π

23π

π

43π

32π

2π

0

1

1

1

1

1

1

1

1

1

1/2

0

1/6

0

0

0

1/6

0

2

1/2

0

1/2

0

1/2

0

1/2

0

3

1

0

2/3

1

0

1

2/3

0

Table 1: Total detection probability 1−S∞ for a quantum walker
on a benzene
ring, for different localized
starting points |ψ(0)⟩=|x⟩. Measurements are performed at x=0 hence
initial conditions on sites 1 and 2 are equivalent to initial conditions
on 5 and 4 respectively. Values of the parameter
γτ are listed in
the first row, and 0<γτ<2π∗ implies all values of
γτ in the interval,
besides the listed special cases, e.g. γτ=π.

v.1.2 Half Dark states

Another peculiar behavior is found if the detection is at
the origin ^D=|0⟩⟨0| and the starting point is
|i⟩ with i=1,2,4,5.
The total probability of detection is found to be, by the method explained in Appendix B,

1−S∞=1/2

(55)

for all
values of 0<γτ<2π besides exceptional
points which are listed in Table
1.
The exceptions include the case when τ is the
full revival time, for which case the probability of being detected
is of course 0.
The behaviour Eq. (55) was observed in
Dhar (); Dhar1 () for even larger systems.
It is remarkable that for certain initial conditions, the detection
of the particle is not guaranteed, and only in half of the measurement
processes we detect the particle, hence we call these initial
conditions
half dark
states.

v.1.3 Starting on site 3 measuring on 0

In contrast, if the starting state is |3⟩
the total probability of detection is found to be 1, if the measurement
time τ is not the full revival time γτ=2π,
or one of the exceptional
sampling times listed in Table
1.
In Appendix B,
we find

⟨n⟩=49+98−8cosγτ+136cos2γτ−19cosγτ+1772(1+cosγτ)

(56)

an equation valid for all γτ besides the exceptional points.
The general behavior of ⟨n⟩
is obviously quite different from the case when the
initial location is 0, compare Fig.
2 and Fig. 3 indicating that the initial
condition plays a crucial rule.
As shown in Fig.
3
the average ⟨n⟩
exhibits nontrivial behavior as it diverges as it approaches some of the
exceptional points. These singularities are found near those
exceptional
sampling times where the total probability of measurement is not one.
Interestingly the values of ⟨n⟩, conditioned on return, are finite at the exceptional points
themselves.

An analytical calculation
for the exceptional sampling times γτ=2π/3 or 4π/3
finds
⟨n⟩=4/3.
This sampling time is unique since the average ⟨n⟩ exhibits
a discontinuity: for γτ in the vicinity of 2π/3 and 4π/3
we
find using
Eq. (56)
⟨n⟩=2 (so at these points the equation is not valid).
Similar to any discontinuity at a point,
the discontinuity of ⟨n⟩ at γτ=2π/3,4π/3
might not be detectable in experiment. However one finds
critical slowing down, namely
the convergence of ⟨n⟩ for any point in the vicinity
of these exceptional points is very slow,
as demonstrated in Fig.
3.

Figure 3:
The average ⟨n⟩ versus γτ Eq. (56).
When
τ→0 we find
⟨n⟩→∞ due to the Zeno effect, another
expected
divergence of ⟨n⟩ is found when the sampling time is the full
revival
time γτ=2π. In addition to these two points we find singularities
also for π/2,π,3π/2.
Notice the discontinuities of ⟨n⟩
for γτ=2π/3 and
γτ=4π/3 which are discussed in the text.
Here |ψ(0)⟩=|3⟩ and the detection is on the origin.
The plot of ⟨n⟩ for this choice of initial condition
is vastly different from that presented in
Fig.
2.

v.2 Rings of Size L

While the benzene ring is instructive, one must wonder how general
are the main results. In Appendix B
we derive the following
four results:

For a ring of size L and for a particle initially on
site x=0 where the measurements are performed, the particle is
detected with probability unity, and in this sense the motion is
recurrent. We emphasize that this result
is a property of the specified initial condition.

For the same initial condition, besides those isolated exceptional
sampling times τ listed below, the average number of
detection events is

⟨n⟩=⎧⎪
⎪
⎪⎨⎪
⎪
⎪⎩L+22L is evenL+12L is odd.

(57)

This result is remarkable since the average is independent of the sampling time τ.
Here we see that ⟨n⟩ is the number of distinct energy
levels of the system.
In the language of Grunbaum () it is the winding
number of the Schur function, or the effective dimension of the Hilbert space.
For large systems ⟨n⟩ grows linearly with the size
of the system L, while from classical random walk
theory we naively expect diffusive
scaling ⟨n⟩∼L2. In that sense the quantum
walk is more efficient (see below further remarks).

The exceptional sampling times τ are given by the rule

ΔEτ=2πn

(58)

where n is an integer, and ΔE=Ei−Ej>0 is the energy
difference between pairs of eigenenergies of the underlying Hamiltonian
H. For example the stationary energies of the benzene ring
are {−2γ,−γ,γ,2γ} as mentioned,
and hence Eq.
(58) predicts the exceptional sampling times
0,π/2γ,2π/3γ,π/γ,....
The condition Eq. (58) implies a partial revival of the wave packet
free of measurement, namely two modes of the system
behave identically when strobed at the period τ.
On these exceptional points the solution exhibits non-analytical behavior.
This is manifested in discontinuities or diverging behavior of ⟨n⟩
or the fluctuations of n and also slow critical-like convergence
to the asymptotic theory. The
exact nature of the non-analytical behavior depends on the initial
condition as we have demonstrated for the benzene ring.

For a particle starting on |0⟩
every time the condition
(58) is met by a pair of energy levels
we reduce the value of
⟨n⟩ by unity.
Thus Eq. (58) is the upper limit of ⟨n⟩ for a system
of fixed size L. More specifically we find that

⟨n⟩=number of distinct phases exp(−iEkτ)

(59)

where Ek are the energy levels of the system.
For nearly any τ this is the same as the number of distinct
energy levels,
but of course for special sampling times, this integer is less
than that.

v.3 Bose-Einstein Distribution

A curiosity is the fact that we may express the solution for the 0 (starting point) to